The terms in the equation (REF ) affect the NEC condition of the time-evolved states significantly in the leading order. If this expression sandwiched between \(\mathinner {\langle {\psi (t_{1})}|}\)  and \(\mathinner {|{\psi (t_{1})}\rangle }\)  is negative, and its magnitude is more than \(\mathinner {\langle {\psi (t_{1})}|}:\hat{T}_{\mu \nu }(t_{2},\vec{x}):n^{\mu }(t_{2},\vec{x})n^{\nu }(t_{2},\vec{x})\mathinner {|{\psi (t_{1})}\rangle }\) , then the \(\mathinner {\langle {\psi (t_{2})}|}:\hat{T}_{\mu \nu }(t_{2},\vec{x}):n^{\mu }(t_{2},\vec{x})n^{\nu }(t_{2},\vec{x})\mathinner {|{\psi (t_{2})}\rangle }<0\)  in the leading order quantum correction. The expressions in (REF ) and (REF ) are made out of the solutions of the Klein-Gordon equations. Since the term in the first parenthesis of (REF ) sandwiched between \(\mathinner {\langle {\psi (t_{1})}|}\)  and \(\mathinner {|{\psi (t_{1})}\rangle }\)  and the expression (REF ) are cubic and quadratic in \(\lbrace \mathcal {O}_{i}^{(k)}\rbrace _{k=1}^{m}\)  respectively, hence, considering a state such that \(\lbrace |\mathcal {O}_{i}^{(k)}|\rightarrow 0\rbrace _{k=1}^{m}\) , we can effectively neglect the effect of these terms compared to the terms in the second parenthesis of (REF ) sandwiched between \(\mathinner {\langle {\psi (t_{1})}|}\)  and \(\mathinner {|{\psi (t_{1})}\rangle }\) . On the other hand, there exist spacetimes [1]}, [2]}, [3]}, [4]}, [5]}, [6]} in which the mode solutions of the Klein-Gordon equation are oscillating in nature. As a result, we can always find a spacetime point in this class of spacetimes at which \(\mathinner {\langle {\psi (t_{2})}|}:\hat{T}_{\mu \nu }(t_{2},\vec{x}):n^{\mu }(t_{2},\vec{x})n^{\nu }(t_{2},\vec{x})\mathinner {|{\psi (t_{2})}\rangle }\)  becomes negative in the leading order quantum correction due to the oscillating nature of the terms in the second parenthesis (REF ). Moreover, as \(\lbrace \mathcal {O}_{i}^{(k)}\rbrace _{k=1}^{m}\)  are the state-dependent free parameters, it is possible to choose such parameters or in other words, construct a quantum state suitably which can violate NEC after time evolution. This clearly shows that the time-evolution can map a NEC satisfying quantum state to a NEC violating quantum state.
